All analogies are imperfect, as is mine, though I would hesitate to say that it could be considered to be one of the worst. I would agree with other opinions which point out that it's more of a co-operative club run by and for the benefit of the members.
I disagree with your point about the the UK not asking to ship things to Poland or Slovakia or France. This is just plain wrong. Of course we are asking to do these things. In fact I have personal experience of this, as we actually do ship things from the UK to Northern Ireland, the Republic of Ireland and the EU. I now have to fill out custom forms and get approval before I can do this. Most bizarrely, I have to do it if I want to ship from where we are in Liverpool (in the UK) to Belfast (also in the UK). I have to get a permit to go into the county of Kent if I want to send a vehicle through the Channel Tunnel to France (£300 fine if I don't). I sometimes think that I have fallen into a dystopian parallel universe.
You are correct that NI and the Republic have a long and ingrained relationship, but the separation between them was largely removed only quite recently by the Good Friday Agreement which has brought relative peace to the region. It has everything to do with the EU because the EU were sensible enough to not countenance the re-imposition of a border between Northern and Southern Ireland and so an agreement (the Northern Ireland Protocol) was agreed between the EU and the UK which puts the border relatively speaking in the Irish Sea. So Northern Ireland is still effectively in the EU customs union and single market. If you think that's complicated, we haven't even started yet :) I'm not surprised you're tired.
As far as the EU is concerned, I don't think that many people in mainland Europe really care that much. They have much more interesting things to do. Those that do care think we're a bit stupid. The EU project will carry on quite happily without the UK. I don't see much evidence of rejoicing on the part of Europeans. Also, quite a few Europeans don't have a terribly high opinion of the EU and certainly don't see it as a Nirvana; and the UK will not be marching to hell. Things are getting more expensive and some opportunities have diminished, but it's not an Armageddon as some would have you believe. Leaving the EU was rather pointless, that's all :|